| If \[5\sin \theta +12\cos \theta =13,\] then what is\[5\cos \theta -12\sin \theta \]equal to? |
A) \[-\,\,2\]
B) \[-\,\,1\]
C) \[0\]
D) \[1\]
Correct Answer: C
Solution :
| \[\because \]\[5\sin \theta +12\cos \theta =13\] |
| On squaring both sides, we get |
| \[25{{\sin }^{2}}\theta +144{{\cos }^{2}}\theta +120\sin \theta \cos \theta =169\] |
| \[\Rightarrow \]\[25\,\,(1-{{\cos }^{2}}\theta )+144\,\,(1-{{\sin }^{2}}\theta )\] |
| \[+\,\,120\sin \theta \cos \theta =169\] |
| \[\Rightarrow \]\[25-25{{\cos }^{2}}\theta +144-144{{\sin }^{2}}\theta \] |
| \[+\,\,120\sin \theta \cos \theta =169\] |
| \[\Rightarrow \]\[25{{\cos }^{2}}\theta +144{{\sin }^{2}}\theta -120\sin \theta \cos \theta =169-169\] |
| \[\Rightarrow \]\[{{(5\cos \theta -12\sin \theta )}^{2}}=0\] |
| \[\Rightarrow \]\[5\cos \theta -12\sin \theta =0\] |
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